Geometric & Functional Analysis GAFA

, Volume 9, Issue 1, pp 1–28 | Cite as

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

  • P. Albuquerque

Abstract.

Let X = G/K be a Riemannian symmetric space of noncompact type and \( \Gamma \) a discrete “generic” subgroup of G with critical exponent \( \delta(\Gamma) \). Denote by \( X_{reg} (\infty) \) the set of regular elements of the geometric boundary \( X(\infty) \) of X. We show that the support of all \( \Gamma \)-invariant conformal densities of dimension \( \delta(\Gamma) \) on \( X_{reg} (\infty) \) (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on \( X(\infty) \). This provides information on the large-scale growth of \( \Gamma \)-orbits in X. If in addition we assume \( \Gamma \) to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine \( \delta(\Gamma) \) and this G-orbit for lattices, and show that they are of divergence type.

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • P. Albuquerque
    • 1
  1. 1.CUI, Université de Genève, 24 rue de Général-Dufour, CH-1211 Genève 4, e-mail: Paul.Albuquerque@cui.unig.chCH

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